Why is the Legendre symbol $(\frac a p)$ not defined if $p | a$ in some books ?

In some textbooks I've come to notice that the legendre symbol $(\frac a p)$ is not defined if $p | a$, where $p$ is a prime and $a$ an integer.

However, on Wikipedia http://en.wikipedia.org/wiki/Legendre_symbol the legendre symbol is defined in the case $p | a$ with value $(\frac a p) = 0$.

I've looked at some theorems to see why the authors doesn't define the legendre symbol in the case $p|a$, but as I see it most results are perfectly valid with the "extended" definition ?

  • 1
    $\begingroup$ Because we are working in the multiplicative group. $\endgroup$ – André Nicolas May 20 '14 at 18:59
  • $\begingroup$ So we only want numbers with inverses modulo $p$ ? But why does some books allow the other case also ? $\endgroup$ – user141901 May 20 '14 at 19:10
  • $\begingroup$ Because it really doesn't matter whether we assign a value to $(a/p)$ when $p$ divides $a$, or what value (as long as it is not $\pm 1$) since we will never actually use it. $\endgroup$ – André Nicolas May 20 '14 at 19:32
  • $\begingroup$ Can you give an example why it may not be assigned $\pm 1$ when $p | a$ ? Also can you give a little context why it doesn't matter whether or not to assign a value to the symbol when $p$ divides $a$ ? $\endgroup$ – user141901 May 21 '14 at 6:15
  • 1
    $\begingroup$ Well, one useful thing is $(a/p)(b/p)=(ab/p)$, If we set $(p/p)=1$ we lose that. $\endgroup$ – André Nicolas May 21 '14 at 12:34

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy